It is shown that if A, B,X, and Y are operators acting on a finite dimensional Hilbert space, then. ωu (AXB∗ ± BYA∗) ≤ 2 ∥A∥ ∥B∥ ωu ([0 X, Y 0]) where ωu (T ), ∥T ∥, are, respectively, the U-numerical radius, the spectral norm, of an operator T .
This article presents a tool for international market selection (IMS) that integrates Neutrosophic Analytic Hierarchy Process (Neutrosophic AHP) and Monte Carlo simulation to reduce uncertainty in export decision-making. The methodology begins with a comprehensive literature review identifying five key criteria and twenty-three sub-criteria for IMS, supported by the insights of five notable authors in the field. Using Neutrosophic AHP, the weights of each criterion and sub-criterion are calculated and incorporated into a mathematical model designed for market selection. Data are collected from globally renowned sources and adjusted to probability distributions, enabling scenario simulation through Monte Carlo. The developed algorithm evaluates 193 countries, generating a ranking of potential destinations based on the determined weights and obtained information. The tool is validated by testing hundreds of products from 4,290 tariff lines under the SA 2012 version, confirming its applicability across diverse commercial contexts. The results highlight the tool's ability to accurately and adaptively identify viable export markets, offering a robust model for strategic decision-making in business internationalisation.
In this paper, we apply neutrosophic N-structures in ternary semirings. We consider ternary neutrosophic N-subsemirings of ternary semirings. We investigate the conditions for neutrosophic N-structures to be neutrosophic ternary N-subsemirings. In addition, we show the relation between ternary subsemirins and neutrosophic ternary N-subsemirins. Finally, we showed that the homomorphic preimage of the neutrosophic ternary N-subsemirings is a neutrosophic ternary N-subsemirings and the onto homomorphic image of the neutrosophic ternary N-subsemiring is also a neutrosophic ternary N-subsemirings.
In this paper, we study the relationships between the Neutrosophic quasi-Frobenius rings and the Neutrosophic of local rings and Artinian rings. In addition, we present study the relationship between the Neutrosophic quasi-Frobenius ring and some concepts such as Neutrosophic semisimple ring, Neutrosophic module injective and Neutrosophic Noetherian ring. Finally, we introduce some mathematical formulas with an commutative, coherent and Neutrosophic perfect ring, through which we obtain the Neutrosophic quasi-Frobenius ring.
In this paper we apply the neutrosophic set on the concept of the UP-algebra to obtain some types of neurosophic sets satisfies certain conditions which are called neutrosophic Up-subalgebras. Several types of these neutrosophic Up-subalgebras are introduced and their properties are investigated. Also, illustrative examples are given when they are needed.
This paper explores the fundamental concepts of sub-level subgroups, element orders, normalizers, and centralizers within the framework of neutrosophic group theory. Additionally, it examines quotient groups and the index of a subgroup, extending classical algebraic structures to a neutrosophic setting. Finally, a generalized formulation of Lagrange’s theorem is presented, demonstrating its applicability in the neutrosophic environment and highlighting its implications for uncertain and indeterminate group structures.
In this paper, we study the notion of principal (crisp) fuzzy ideals (resp. filters) on the setting of trellises (or weakly associative lattices as called by several authors). More specifically, we introduce the notions of L-fuzzy ideals and L-fuzzy filters on a given trellis and provide basic characterizations of these notions based on their weakly associative meet and join operations. We pay particular attention to the kind of principal L-fuzzy ideals (resp. filters) on a given trellis, which are more complicated in the absence of the (associativity) transitivity property.
The aim of this article is proposed the notion of Correlation Measure for Neutrosophic Filter (NF). Additionally using correlation measure for neutrosophic set the application of medical diagnosis were discussed with numerical example.
Ranking algorithms are very important tools in decision making. There are two ranking algorithms for nvalued neutrosophic tuplets (Single-Valued MultiNeutrosophic tuplets): The S-ranking algorithm of Single- Valued MultiNeutrosophic tuplets, which is introduced by F.Smarandache in 2023, and the N-ranking algorithm of n-valued neutrosophic tuplets, Which is introduced by V. L. Nayagam and Bharanidharan R. in 2023. In this paper we show (by examples) that these two ranking algorithms are not a total ordering for the set of n-valued neutrosophic tuplets. These algorithms do not taking into account the number of sources, which is a very important factor in neutrosophic n-valued refined sets theory. We introduce two ranking algorithms: The integrated S-ranking algorithm of Single-Valued MultiNeutrosophic tuplets, and the integrated N-ranking algorithm of n-valued neutrosophic tuplets. These algorithms are improvements of the S-ranking algorithm of Single-Valued MultiNeutrosophic tuplets, and the N-ranking algorithm of n-valued neutrosophic tuplets, respectively, and taking the number of sources into account. We construct different examples to show that each step in the integrated ranking algorithms is necessary to make them a total ordering for the set of all n-valued neutrosophic tuplets.
Neutrosophic cognitive maps are expansion of fuzzy cognitive maps, containing indetermination in causal relations. Fuzzy cognitive maps do not require an indeterminate relationship, making it less adequate for real-time applications. A logic in which every proposition is projected to have the truth percentage in subset T and the falsity percentage in subset F is named Neutrosophic Logic. This logic is also considered the general form of Intuitionistic fuzzy logic. Stock price prediction is a main topic in economics and finance, which has promoted the priority of investigators in recent years to improve improved predictive methods. Predicting price and tendency of the stock market denote indispensable features of finance and investment. Many scientists have presented their ideas to predict the market price to make money while trading utilizing different methods like statistical and technical analysis. This manuscript proposes a Neutrosophic Cognitive Map-Based Short-Term Financial Stock Market Price Trend Prediction (NCM-SFSMPTP) model. The main goal of NCM-SFSMPTP technique relies on improving the accurate approach for stock market price trend prediction. At first, the min-max normalization methodology is utilized in the data normalization phase to standardize and scale data for consistency, comparability, and efficient processing. For the classification process, the neutrosophic cognitive map (NCM) technique is employed. Finally, the improved arithmetic optimization algorithm (IAOA)-based hyper-parameter selection is implemented to enhance the classification outcomes of the NCM system. The performance validation of the NCM-SFSMPTP methodology is verified under the Apple Stock Price Trend and Indicators dataset and the outcomes are determined regarding to several measures. The experimental validation of the NCM-SFSMPTP method illustrated a superior accuracy value of 94.79% over existing models in stock market price trend prediction process.
The first appearance of COVID-19 in late 2019 and spread rapidly throughout the world until it became a global pandemic, and the World Health Organization announced some vaccines, and the emergence of a mutated version of COVID-19 was reported in several countries, including Iraq, and we will take care of conducting a study on the spread and dynamics of a virus, this work will be based on the study of the dynamics 3D harvesting predator (COVID-19) differential-algebraic predator-prey economic model (DA-PPM) with functional responses of Holing type-II. The appropriate and realistic description with high accuracy of this phenomenon, which may be natural and emerging as such models, has proven the sentimentality and existence of the solution to the system, and the stability of the system, was discussed in a manner similar to the stability of Matignon. The numerical results showed that the variables of stable unhappy situations have an effect, and this important study can be used as one of the methods of health science to control the spread of COVID-19 and its advanced models. One of the critical aspects of sustainable development is building resilient health systems capable of dealing with epidemics and other crises, the mathematical model (DA-PPM) was applied to analyze the sustainability of health systems under the pressure of Covid-19 and evaluate how long-term public health policies and interventions can prevent overexploitation of resources. Ensuring equitable access to care. The application of the mathematical model to understand the spread of the epidemic is discussed to observe the spread of the epidemic, the possibility of coexistence with it, its close relationship with sustainable development, and to emphasize the importance of the flexibility of the health system. In addition, we apply our results on the neutrosophic supposed data that deals with uncertainty in real-life measurements and compare it with the classical results.
Accurate and timely recognition of plant diseases is crucial to prevent crop loss and ensure global food security. This paper presents a robust ensemble-based framework that combines six classical and state-of-the-art deep convolutional neural networks (DCNNs), including a ConvNeXt architecture, and integrates Neutrosophic Science to better handle uncertainty in leaf images. The proposed approach features three main components: (1) transfer learning with pre-trained DCNNs, (2) a model-averaging strategy to stabilise individual predictions, and (3) a stacked ensemble design that employs a radial basis function (RBF) meta-learner to refine the classification outputs. Experiments on the Plant Village dataset, comprising 54,305 segmented images of 38 plant diseases, included 10-fold cross-validation. The results show that the final stacking ensemble achieved near-perfect performance with 99.97% accuracy and an F1 score of 99.55% on an unseen test set of 27,160 images. Compared with standalone models, the ensemble demonstrated greater robustness in distinguishing visually similar diseases, benefiting from the complementary strengths of multiple DCNN architectures. The Neutrosophic component further enhances reliability by modelling uncertainties due to noise, occlusions, and environmental variations. Although a higher computational overhead and modest misclassifications remain, especially in certain visually overlapping classes, this study demonstrates the effectiveness of an ensembledriven, uncertainty-aware strategy. These findings hold considerable promise for real-world agricultural applications, where rapid and accurate disease diagnosis is paramount.
Regression modeling is a significant statistical tool aimed at quantifying and understanding the nature of relations between the predictor and response variables. The routine parameter estimation procedures, like OLS and ML, are based heavily on the assumption of normality in data, which will not be the case for most real-world data scenarios. The paper presents a Neutrosophic approach for the estimation of parameters in multiple linear regression models, making use of the Neutrosophic principles to treat uncertainties, indeterminacies, and inconsistencies in actual data, a proposed method is called the Simple Averaging Method, or SAM. This is a robust alternative to traditional methods and provides reliable results even if the assumptions of normality are not held. SAM performance is tested using real-time crime data in the USA and demonstrates its capabilities to deal with complex datasets. The comparative analysis between the OLS model and the same model is done via RMSE and MAD metrics. The results show that SAM significantly outperforms OLS with an RMSE of 34.37598 in contrast to 58.05248 for OLS. Graphical analysis further confirms SAM's performance over and above OLS. Critical issues of regression modeling with incorporation of neutrosophic logic cover their critical challenges, especially when standard assumptions are violated.
Fuzzy fractional partial differential equations have become a powerful approach to handle uncertainty or imprecision in real-world modeling problems. In this article, two compact finite difference schemes, the compact Crank-Nicolson and the compact center time center space methods, were developed and used to obtain a numerical solution for fuzzy time fractional wave equations in the double parametric form. The principles of fuzzy set theory are utilized to perform a fuzzy analysis and formulate the proposed numerical schemes. The Caputo formula is used to define the time-fractional derivative considered. The stability of the proposed schemes is analyzed by means of the Von Neumann method. To illustrate the practicality of the numerical methods, a specific numerical instance was performed. The outcomes were showcased through tables and figures, revealing the efficacy of the schemes in terms of accuracy and their ability to decrease computational expenses.
The neutrosophic set, a mathematical framework that accounts for truth, indeterminacy, and falsity, plays a crucial role in enhancing artificial intelligence (AI)-driven language education. By integrating neutrosophic logic, AI systems can better handle linguistic ambiguities, dynamically adapt learning materials, and offer more precise and personalized feedback. This paper explores the application of neutrosophic theory in intelligent tutoring systems (ITS), natural language processing (NLP), and AI-assisted feedback mechanisms, all within an uncertainty-based framework. Through the incorporation of neutrosophic models, AI can more effectively assess learner responses by considering elements of truth, uncertainty, and falsehood, leading to more adaptive and context-aware language instruction. Furthermore, the study highlights how AI, powered by neutrosophic logic, contributes to breaking language barriers, increasing accessibility, and fostering inclusive learning environments. Ethical concerns, bias mitigation, and data privacy challenges in AI-driven language learning are also addressed, emphasizing the need for responsible AI implementation. Finally, the paper underscores the synergistic balance between AI and human educators, advocating for adaptive AI frameworks that enhance linguistic comprehension while ensuring pedagogical integrity. Future research directions focus on leveraging neutrosophic logic to further improve AI's reliability, adaptability, and overall effectiveness in personalized language education.
In this paper, we defined some new kinds of continuous functions in fuzzy neutrosophic topology and called fuzzy neutrosophic - continuous, fuzzy neutrosophic weakly continuous, fuzzy neutrosophic strongly - continuous, fuzzy neutrosophic -contra continuous, fuzzy neutrosophic weakly -contra continuous and fuzzy neutrosophic strongly -contra continuous functions. Then, we defined the relationship between the define functions with their comparative.
Assume R is a commutative ring with unity. The clean graph CL(R) is defined in which every vertex has the form (a, v), where a is an idempotent in R and v is a unit. In CL(R), two distinct vertices (a1, v1) and (a2, v2) are adjacent if a1a2 = a2a1 = 0 or v1v2 = v2v1 = 1. In this paper, we show that the clean graph CL(R) over a ring of order p2 can be defined only if R is one of the rings: Zp2 ,Zp ⊕Zp,Zp(+)Zp and GF(p2). Then, we study the spectrum, the biclique partition number, and the eigensharp property for the these clean graphs.
Background: Neutrosophy is the subject area of philosophy that researches all associated with neutralities, owing to the contradictory information, lack of information, imprecise and paradoxical information, among them. The scale's design is organized to take the subjective quality of opinion, being responsible for either uncertainty or the indeterminacy of the respondents' opinions. It relies on the triple refined indeterminate neutrosophic sets (NS) for improved accuracy in understanding the agreement or disagreement level on particular items, like the competence of activities cost and financial management inside the legal services. Currently, customer abrasion is more and more serious in commercial banks, mainly, high-valued customers in retail banking. Therefore, it is stimulated to advance a prediction mechanism and recognize this customer may be at attrition risk. Thus, recognizing and lowering customer churn has become important for financial institutions trying to maintain customers. Currently, several researchers concentrate on customer attrition rate studies utilizing sophisticated machine learning (ML) and deep learning (DL) methods. Methodology: This study develops a Customer Attrition Prediction Using Triple Refined Indeterminate Neutrosophic Sets with an Optimization Algorithm (CAP-TRINSOA) technique. The main aim of the CAP-TRINSOA technique is to improve the attrition prediction of a customer in large-scale financial sectors using state-of-the-art techniques. In the initial stage, the data normalization employs mean normalization to transfer input data into an even format. Furthermore, the classification process is performed by implementing the triple refined indeterminate neutrosophic sets (TRINS). Finally, the honey badger algorithm (HBA) alters the parameter tuning value of the TRINS method optimally and results in greater performance of classification. Results: An extensive set of simulations is accomplished to exhibit the promising results of the CAP-TRINSOA method under the bank customer churn prediction dataset. The experimental validation of the CAP-TRINSOA technique portrayed a superior accuracy value of 97.65% over exisitng model in the customer attrition prediction process.
A neutrosophic set (NS) contains 3 modules such as the degree of truth (T), degree of falsity (F), and degree of indeterminacy (I). While fuzzy graphs (FG) occasionally fall short of providing optimum outcomes, the NS and neutrosophic graphs (NG) provide a strong substitute, which efficiently handles the uncertainties related to indeterminate and inconsistent data in real-life scenarios. Conversely, bipolar neutrosophic methods, which account for both negative and positive effects, deliver a more flexible and applicable technique. Financial crisis prediction (FCP) is inherent in the detection of major social and economic impacts that crises of financial might hold on a global measure. It generally outcomes in vast financial losses, redundancy, and losses in values of assets that lead to significantly affected individuals and businesses. In recent times, the credit risk prediction methods have aided businesses in resolving whether to award credit to users who applied. This paper presents the Financial Credit Risk Forecasting Using Bipolar Single-Valued Neutrosophic Graph Sets Approach (FCRF-BSVNGSA) method. The main intention of the FCRF-BSVNGSA method is to develop an effective method for financial credit risk prediction using advanced methods. At first, the data normalization stage utilizes Z-score normalization for converting the input data into a beneficial format. Furthermore, for the financial credit risk classification process, the proposed FCRF-BSVNGSA model employs the bipolar single-valued neutrosophic graphs (BSVNG) approach. Finally, the multi‐objective hippopotamus optimization (MOHO) approach fine-tunes the hyperparameter values of the BSVNG model optimally and results in superior classification performance. An extensive simulation of the FCRF-BSVNGSA approach is performed under the Statlog (German Credit Data) dataset. The experimental validation of the FCRF-BSVNGSA approach portrayed a superior accuracy value of 95.59% over exisitng techniques.
